A second-order topological phase protected by rotation symmetry has \((d-2)\)  dimensional anomalous boundary states, for example isolated Majorana bound states at the corners of a two-dimensional crystal or chiral/helical modes at the hinges of a three-dimensional crystal.
This is illustrated in Figs. REF (a) and (b).
We require that the anomalous boundary excitations are intrinsic [1]}, [2]}, [3]}, i.e., we allow any changes of the crystal termination consistent with the rotation symmetry, for instance a decoration of the boundary with lower-dimensional topological phases.
This property ensures that the anomalous boundary excitations are truly attributed to the topology of the \(d\) -dimensional bulk.
Furthermore, throughout this article we focus on tenfold-way anomalous boundary states appearing in systems described by quadratic fermionic Hamiltonians.
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